# Simulating the path for Interest Rate

There are many ways to short term rates like Ho-lee process, HW process. However I failed to understand how this information can be used to simulate the process for Overnight rate like EONIA etc.

Can you please refer to any online technical papers to simulate such Overnight process?

Just to explain more, let say I define a HW process as follows

$$dr(t) = (\theta(t)-\alpha r(t))dt+\sigma dW_t$$

With this process, I can estimate a discount bond as $$P(t, t+1)$$ based on the estimated parameters (refer to https://en.wikipedia.org/wiki/Hull–White_model)

So is it right to day that the process for OI rate is just the process for $$P(t,t+1)$$?

Is there any software implementation like R/Python that someone can please refer to?

Thank you very much.

• It is very common to associate a short rate with the overnight rate like EONIA as it is, after all, very short -- 1 day. So any paper/implementation for a short rate model can be used as is. Commented Mar 17, 2021 at 18:58
• @piterbarg - I have modified my original post with more information. Do you think I am going to right direction? Commented Mar 17, 2021 at 20:34

You are on the right path but here $$P(t,t+1)$$ is not your overnight rate but a daily discount factor. To convert to a simply compounded daily rate, which would be your overnight rate, you would do something like this $$R(t,t+1) = (1 - P(t,t+1))/(\tau P(t,t+1))$$
Here $$\tau$$ is your daycount fraction for $$1$$ day
As I mentioned in the comment $$R(t,t+1)$$ is very close to the short rate $$r(t)$$ and, depending on the intended application, you might want to use the latter as it is a bit simpler. Otherwise use $$R$$